Fundamental Facts

A fraction represents a part of a whole. It is expressed as \( \frac{\text{Numerator}}{\text{Denominator}} \), where the denominator is not zero. To write a fraction correctly, all parts counted must be equal in size.

Introduction to Fractions

A fraction is a number that denotes a part of a whole or a group. It is written as \( \frac{a}{b} \) where \( b \neq 0 \). Fractions can be represented on a number line by dividing the segment between 0 and 1 into \( b \) equal parts and locating the point at \( a \) parts.

Types of fractions include:

• Proper fraction: Numerator less than denominator, e.g., \( \frac{1}{3} \).
• Improper fraction: Numerator greater than or equal to denominator, e.g., \( \frac{5}{3} \).
• Mixed fraction: Combination of a whole number and a proper fraction, e.g., \( 3 \frac{4}{5} \).

Equivalent fractions are fractions that represent the same value, found by multiplying or dividing numerator and denominator by the same non-zero number.

A fraction is in simplest form if numerator and denominator have no common factors other than 1.

Fractions

A fraction \( \frac{3}{5} \) means the whole is divided into 5 equal parts and 3 parts are taken. Examples include half \( \frac{1}{2} \) and quarter \( \frac{1}{4} \).

Example 1

Write the fraction of the shaded portions.

First circle: 5 equal parts, 4 shaded, fraction \( \frac{4}{5} \).

Second circle: 2 equal parts, 1 shaded, fraction \( \frac{1}{2} \).

Unit fractions are fractions with numerator 1, e.g., \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \).

Keywords

• Fraction: part of a whole
• Numerator: number of parts taken
• Denominator: total equal parts

Fractions on the Number Line

To represent \( \frac{4}{5} \) on a number line, divide the segment from 0 to 1 into 5 equal parts and mark the point at 4 parts.

Proper, Improper and Mixed Fractions

Proper fraction: Numerator less than denominator, e.g., \( \frac{1}{3} \).

Improper fraction: Numerator greater than or equal to denominator, e.g., \( \frac{5}{3} \).

Mixed fraction: Whole number plus proper fraction, e.g., \( 3 \frac{4}{5} \).

Conversion formulas:

Improper fraction to mixed number:

a. Divide numerator by denominator: \( \text{Quotient} = q, \text{Remainder} = r \)

b. Write as \( q + \frac{r}{\text{denominator}} \)

Mixed number to improper fraction:

\[ \text{Improper fraction} = \frac{\text{denominator} \times \text{whole number} + \text{numerator}}{\text{denominator}} \]

Keywords

• Proper fraction: numerator < denominator
• Improper fraction: numerator ≥ denominator
• Mixed fraction: whole number + fraction

Equivalent Fractions

Fractions representing the same part of a whole are equivalent. For example, \( \frac{1}{4} = \frac{2}{8} = \frac{3}{12} = \frac{4}{16} \).

To find equivalent fractions, multiply or divide numerator and denominator by the same non-zero number.

Example: Find 5 equivalent fractions for \( \frac{11}{15} \):

\[ \frac{11 \times 2}{15 \times 2} = \frac{22}{30}, \frac{11 \times 3}{15 \times 3} = \frac{33}{45}, \frac{11 \times 5}{15 \times 5} = \frac{55}{75}, \frac{11 \times 10}{15 \times 10} = \frac{110}{150}, \frac{11 \times 20}{15 \times 20} = \frac{220}{300} \]

Simplest Form of a Fraction

A fraction is in simplest form if the highest common factor (HCF) of numerator and denominator is 1.

Example: Simplify \( \frac{36}{24} \)

HCF of 36 and 24 is 12.

Divide numerator and denominator by 12:

\[ \frac{36}{24} = \frac{36 \div 12}{24 \div 12} = \frac{3}{2} \]

Any fraction can be simplified by dividing numerator and denominator by their HCF.

Keywords

• Simplest form: numerator and denominator have no common factor except 1
• HCF: highest common factor

Example: Simplify \( \frac{54}{108} \)

HCF of 54 and 108 is 54.

\[ \frac{54}{108} = \frac{54 \div 54}{108 \div 54} = \frac{1}{2} \]

Like and Unlike Fractions

Fractions with the same denominator are called like fractions, e.g., \( \frac{7}{13}, \frac{6}{13} \).

Fractions with different denominators are unlike fractions, e.g., \( \frac{2}{6}, \frac{1}{5} \).

Keywords

• Like fractions: same denominators
• Unlike fractions: different denominators

Comparing Fractions

To compare fractions, determine which is larger or smaller.

Comparing like fractions: Compare numerators directly.

Comparing unlike fractions:

1. Find LCM of denominators to get common denominator.
2. Convert fractions to equivalent fractions with common denominator.
3. Compare numerators.

Example: Compare \( \frac{1}{2} \) and \( \frac{2}{5} \)

LCM of 2 and 5 is 10.

\[ \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} \]

\[ \frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} \]

Since \( 5 > 4 \), \( \frac{1}{2} > \frac{2}{5} \).

Fundamental Facts

If numerators are equal, the fraction with smaller denominator is larger.

Example: \( \frac{3}{5} > \frac{3}{6} \)

Addition and Subtraction of Fractions

Like fractions: Add or subtract numerators, keep denominator same.

Example: \( \frac{1}{7} + \frac{2}{7} + \frac{3}{7} = \frac{1+2+3}{7} = \frac{6}{7} \)

Unlike fractions:

1. Find LCM of denominators.
2. Convert fractions to equivalent fractions with common denominator.
3. Add or subtract numerators.
4. Simplify the result.

Example: Subtract \( \frac{2}{9} \) from \( \frac{3}{10} \)

LCM of 9 and 10 is 90.

\[ \frac{3}{10} = \frac{3 \times 9}{10 \times 9} = \frac{27}{90} \]

\[ \frac{2}{9} = \frac{2 \times 10}{9 \times 10} = \frac{20}{90} \]

Subtract:

\[ \frac{27}{90} - \frac{20}{90} = \frac{27 - 20}{90} = \frac{7}{90} \]

Keywords

• LCM: Least common multiple